Keywords: Stokes equations, stabilized finite element method, iterative substructuring method, balancing Neumann-Neumann preconditioner, conjugate gradient method, indefinite matrix.

This paper shows a simple algorithm for an iterative substructuring method, similar to those for elasticity problems, using a discretized Stokes equation with a P1/P1 element and a penalty stabilization technique [1]. We introduce two simple algorithms. One uses a symmetric and indefinite matrix and the other uses a skewed matrix which is not symmetric but coercive. Owing to the stability term, the solvabilities of the local Dirichlet problem, of the local Neumann problem with preconditioner, and of the coarse space problem are ensured. For the symmetric formulation, the conjugate gradient (CG) method with a balancing preconditioner [2] is used to solve the linear system of the discretized Stokes equations whose matrix is symmetric but indefinite. Our algorithm is simpler than an extension of iterative substructuring method from the elasticity equations to the Stokes equations [3], where a P2/P0-discontinuous pressure element is used and a careful construction of the coarse space is needed to satisfy the supplementary inf-sup condition. When the procedure of the preconditioned CG does not meet a breakdown, the solution is found in the largest Krylov subspace generated from the initial residual and preconditioned coefficient matrix [4]. This solver is much simpler than the CG method in a "benign space" [3], where incompressibility of the velocity is satisfied in a discrete sense. First, we describe the governing equations and the finite element approximation and two formulations with a symmetric indefinite matrix and with an unsymmetric coercive matrix. Second, we introduce a non-overlapping domain decomposition, construct a Schur complement system, and propose a coarse space for the balancing preconditioner. Finally we discuss the direct factorization solver for the local Dirichlet problem of Schur complement system and for the local Neumann problem of a balancing preconditioner. Though the matrix of the symmetric formulation is indefinite, we can use block-LDL^T factorization for the local problems and CG method for the Schur complement system, which has the advantage of using less memory. We introduce a unified factorization procedure to the local Dirichlet and local Neumann problems for saving memory requirements.

1

F. Brezzi, J.J. Douglas, "Stabilized mixed methods for the Stokes problem", Numer. Math., 53, 225-235, 1988. doi:10.1007/BF01395886

2

J. Mandel, "Balancing domain decomposition", Commun. Numer. Methods Eng., 9, 233-241, 1993. doi:10.1002/cnm.1640090307

3

L.F. Pavarino, O.B. Widlund, "Balancing Neumann-Neumann methods for incompressible Stokes equations", Comm. Pure Appl. Math., 55, 302-335, 2002. doi:10.1002/cpa.10020

4

A. Suzuki, "A parallel finite element solver for large-scale 3-D Stokes problem and its application to Earth's mantle convection problem", Doctoral Thesis, Department of Informatics, Kyoto University, 2003.

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